M. S. LeiferBanff (12th February 2007)

Conditional Density Operators in Quantum Information

quant-ph/0606022 Phys. Rev. A 74, 042310(2006) quant-ph/0611233

Classical Conditional Probability

Generally, and can be ANY events in the sample space.

Special Cases:

1. is hypothesis, is observed data - Bayesian Updating

2. and refer to properties of two distinct systems

3. and refer to properties of a system at 2 different times - Stochastic Dynamics

2. and 3. -

(!, S, µ)

Prob(B|A) =µ (A !B)

µ (A)

A,B ! S µ(A) != 0

A B

B

B

B

A

A

A

! = !1 ! !2

Quantum Conditional Probability

In Quantum Theory

Hilbert Space

-

Density operator

What is the analog of ?

Definition should ideally encompass:

1. Conditioning on classical data (measurement-update)

2. Correlations between 2 subsystems w.r.t tensor product

3. Correlations between same system at 2 times (TPCP maps)

4. Correlations between ARBITRARY events, e.g. incompatible observables

AND BE OPERATIONALLY MEANINGFUL!!!!!

!! H

S ! {closed s-spaces of H}

µ! !

Prob(B|A)

Outline

1. Conditional Density Operator

2. Remarks on Conditional Independence

3. Choi-Jamiolkowski Revisited

4. Applications

5. Open Questions

1. Conditional Density Operator

Classical case: Special cases 2. and 3. -

We’ll generally assume s-spaces finite and deal with them by defining integer-valued random variables

abbreviated to

abbreviated to

Marginal

Conditional

! = !1 ! !2

!1 = {X = j}j!{1,2,...N} !2 = {Y = k}k!{1,2,...M}

P (X = j, Y = k) P (X, Y )

P (Y |X) =P (X, Y )P (X)

P (X) =!

Y

P (X, Y )

!

j

f(P (X = j, Y = k))!

X

f(P (X, Y ))

1. Conditional Density Operator

Can easily embed this into QM case 2, i.e. w.r.t. a tensor product

Write

Conditional Density operator

Equivalently

where and is the Moore-Penrose pseudoinverse.

More generally, so how to generalize the conditional density operator?

HX !HY

!XY =!

jk

P (X = j, Y = k) |j! "j|X # |k! "k|Y

!Y |X =!

jk

P (Y = k|X = j) |j! "j|X # |k! "k|Y

!Y |X = !!1X ! IY !XY

!X = TrY (!XY ) !!1X

[!X ! IY , !XY ] "= 0

1. Conditional Density Operator

A “reasonable” family of generalizations is:

Cerf & Adami studied

when well-defined.

Conditional von Neumann entropy

Many people (including me) have studied which we’ll write

Can be characterized as a +ve operator satisfying

!(n)Y |X =

!!! 1

2nX ! IY !

1nXY !

! 12n

X ! IY

"n

!(!)Y |X = lim

n"!!(n)

Y |X

!(!)Y |X = exp (log !XY ! (log !X)" IY )

S(Y |X) = S(X, Y )! S(X) = !Tr!!XY log !(!)

Y |X

"

!(1)Y |X !Y |X

TrY

!!Y |X

"= Isupp(!X)

2. Remarks on Conditional Independence

For the natural definition of conditional independence is entropic:

Equivalent to equality in Strong Subadditivity:

Ruskai showed it is equivalent to

Hayden et. al. showed it is equivalent to

Surprisingly, it is also equivalent to

!(!)Y |X

S(X : Y |Z) = 0

S(X, Z) + S(Y, Z) ! S(X, Y, Z) + S(Z)

!(!)Y |XZ = !(!)

Y |Z ! IX

!XY Z =!

j

pj"XjZ(1)j! #

XjZ(2)j

HXY Z =!

j

HXjZ(1)

j!H

YjZ(1)j

!XY |Z = !X|Z!Y |Z !XY Z = !XZ!!1Z !Y Z

2. Remarks on Conditional Independence

But this is not the “natural” definition of conditional independence for

and are strictly weaker and inequivalent to each other.

Open questions:

Is there a hierarchy of conditional independence relations for different values of ?

Do these have any operational significance in general?

!Y |XZ = !Y |Z ! IX !X|Y Z = !X|Z ! IY

n = 1

n

3. Choi-Jamiolkowsi Revisited

CJ isomorphism is well known. I want to think of it slightly differently, in terms of conditional density operators

Let

direction

direction

!Y |X != EY |X : B(HX)" B(HY )Trace Preserving

Completely PositiveTPCP

!+X!|X =

!

jk

|j! "k|X # |j! "k|X!

EY |X ! !Y |X

!Y |X ! EY |X

!Y |X = IX ! EY |X!

!!+

X!|X

"

EY |X (!X) = TrXX!

!"+

X!|X!X ! "Y |X!

"

3. Choi-Jamiolkowsi Revisited

What does it all mean?

Cases 2. and 3. from the intro are already unified in QM, i.e. both experiments

can be described simply by specifying a joint state .

Do expressions like , where , are POVM elements have any meaning in the time-evolution case?

!XY!=

!!X , !Y |X

" !=!!X , EY |X

"

Prepare !XY Prepare !X

EY |X

Time evolution

!XY

Tr (MX !NY !XY ) MX NY

3. Choi-Jamiolkowski Revisited

Lemma: is an ensemble decomposition of a

density matrix iff there is a POVM s.t.

and

Proof sketch: Set

! =!

j

pj!j

! M =!

M (j)"

pj = Tr!M (j)!

"!j =

!!M (j)!!

Tr!M (j)!

"

M (j) = pj!! 1

2 !j!! 1

2

3. Choi-Jamiolkowski Revisited

-measurement of

Input:

Measurement probabilities:

-preparation of

Input: Generate a classical r.v. with p.d.f

Prepare the corresponding state:

M

!

!

M

M !

!

M !

P (M = j) = Tr!M (j)!

"

P (M = j) = Tr!M (j)!

"

!j =!

!M (j)!!

Tr!M (j)!

"

3. Choi-Jamiolkowski revisited

is the same for any POVMs and .

P (M,N)M N

!XY

X Y

NM

EY |X

!X !TX

MTM

NN

XX

YY

!XY

measurement

measurementmeasurement

measurementmeasurement

preparation

4. Applications

Relations between different concepts/protocols in quantum information, e.g. broadcasting and monogamy of entanglement.

Simplified definition of quantum sufficient statistics.

Quantum State Pooling.

Re-examination of quantum generalizations of Markov chains, Bayesian Networks, etc.

5. Open Questions

Is there a hierarchy of conditional independence relations?

Operational meaning of conditional density operator and conditional independence for general n?

Temporal joint measurements, i.e. ?

The general quantum conditional probability question.

Further applications in quantum information?

Tr (MXY !XY )